Abstract. In the classical k-median problem the goal is to select a subset of at most k facilities in order to minimize the total cost of opened facilities and established connections between clients and opened facilities. We consider thecapacitated version of the problem, where a single facility may only serve a limited number of clients. We construct approximationalgorithms slightly violating the capacities based on rounding a fractional solution to the standard LP.

It is well known that the standard LP has unbounded integrality gap if we only allow violating capacities by a factorsmaller than 2, or if we only allow violating the number of facilities by a factor smaller than 2. In an earlier version of our work we showed that violating capacities by a factor of 2+ε is sufficient to obtain constant factorapproximation of the connection cost. In this paper we substantially extend this result in the following two directions. First, we extend the 2+ε capacity violating algorithm to the more general k-facility location problem with uniform capacities, where opening facilities incurs a location specific opening cost. Second, we show that violating capacities by a slightly bigger factor of 3+ε is sufficient to obtain constant factorapproximation of the connection cost also in the case of the non-uniform hardcapacitatedk-median problem.

Joint work with Jarosław Byrka, Krzysztof Fleszar, and Joachim Spoerhase, to appear at SODA 2015.